It is well known that solutions of the Toda equations correspond to
certain kinds of harmonic maps.
The above relation between solutions \(w\)  and 1-forms \(\eta (z) \, dz\)  is, in fact, an example of the
generalized Weierstrass representation (or DPW representation) for harmonic maps of surfaces into symmetric spaces [1]}. This is based on the loop group Iwasawa factorization [2]}.
